Actually if I remember correctly (at least in NJ) the registration shows the name of the creditor as the owner. That was my only past experience. Mine was a vote in the "no you don't own it" direction and mine was financing not even a lease. (Not that they are fundamentally so different)
Vehicle registration rules in the UK[0] make a clear distinction between the "registered keeper" of a vehicle (the person who "keeps" it, uses it day-to-day, and who is responsible for taxing it), and a vehicle's owner.
A vehicle's registration papers show the keeper's name and details but the document states in large letters "This document is not proof of ownership".
[0] https://www.gov.uk/vehicle-registration/new-and-used-vehicle...
I think our paperwork shows the registration info of the person who registred it. Like in UK this is not ownership by itself. The paperwork also has a section for any liens on the property. This can be the creditor from the beginning or a private person or a government entity can have a claim due to debts. Nowadays these are all in the system of course. The paper card does not mean anything. Everything is looked up on the spot.
Also this depends on the state, some US states are set up where the bank owns the title until the borrower pays it off, and others have the borrower on the title from the start but with a lien on the car by the lender in case the borrower doesn't pay. https://realestatelicensewizard.com/title-theory-vs-lien-the...